Institution:Dept. of
PhysicsUCB 390,
University of ColoradoBoulder,
CO80309-0390

Abstract Title:
Relativity in the Global Positioning System

Body
The Global Positioning System (GPS) provides a superb
opportunity to discuss special and general relativity in an
elementary way, as there are many relativistic effects that
can be related to a few fundamental principles with simple
arguments. The
principles needed include the first and second postulates of
special relativity, particularly the constancy of the speed
of light c, and the weak principle of equivalence. Thought
experiments using the constancy of c lead directly to the
relativity of simultaneity, the first-order Doppler effect,
time dilation, and the Sagnac effect. Then the principle of
equivalence leads to the gravitational effect on clock
frequencies. In the GPS all these effects are large. Also,
GPS data from many receivers is widely available on the
world-wide-web, and can be downloaded and analyzed by anyone
with sufficient patience to develop his or her own
software. This provides opportunities for nontrivial
projects, as well as for new applications to be developed
and tested.

Body Line integral
convolution (LIC) is a technique introduced by Cabral and
Leedom (1993) for visualizing the spatial structure of two
dimensional vector fields at a resolution close to the
resolution of the display. Sundquist (2003) has developed a
way to extend LIC visualization to time dependent vector
fields (DLIC). In particular, he has used the DLIC
technique to make movies of electromagnetic electric dipole
radiation that show the details of the time dependent
structure of that radiation in the near, intermediate, and
far zone, at a resolution close to the resolution of the
display. Examples of these DLICs can be found on the web
through links at
http://jlearn.mit.edu. The purpose of this poster is to
acquaint the GR community with the existence of this
technique, to show examples of it, and to solicit input as
to whether this technique can be used to visualize
gravitational radiation.

BodyUC Davis offers
a "Topics in Physics for Nonscientists" course with an
emphasis that changes from year to year. I describe an
attempt to teach general relativity, using Thorne's Black
Holes and Time Warps as a text and assigning a final essay
rather than an
exam. The goal was for students to learn both some
qualitative features of general relativity -- what does it
mean that spacetime is "four dimensional" and "curved," what
is the principle of equivalence, what is an event horizon,
how might we detect gravitational waves, etc. -- and
something about how theoretical physics works -- how
physicists approach new questions, what kinds of patterns we
look for, how we view the "aesthetics" of physics, what
roles experiment and theory play, etc.

BodyI present an
outline of a course on GR being designed for seniors and
juniors of my department, keeping in mind their varied
backgrounds and abilities. The course will have a
multi-layered and multi-level structure, incorporating
in-class lectures and discussions, reading assignments,
online research, and individual computational projects. I
shall seek ideas and input from participants at the AAPT
workshop.

BodyThere are many reasons to create computer-based material for
relativity. Special and general relativity are full of
(apparent) paradoxes, and, like quantum mechanics, captivate
students’ interest in physics. Because relativity focuses
on abstract concepts, visualization is especially valuable.
This poster will report on the development of new Open
Source Physics (OSP) simulations and curricular material
created for the exploration of relativity. Examples,
including the gravitational red shift and the trajectories
of particles and light rays in the vicinity of non-spinning
and spinning black holes, will be available on CD. OSP is an
NSF-funded curriculum development project that is developing
and distributing a code library, programs, and examples of
computer-based interactive curricular material. The OSP
code library, documentation, and sample curricular material
can be downloaded from
http://www.opensourcephysics.org/apps/gr/index.html.

Partial funding for this work was obtained through NSF grant
DUE-0442581.

Institution:Dept. of
Physics, Univ. of Toronto60 St. George
St.Toronto, ON M5S
1A7 Canada

Abstract Title:
A Voluntary Straitjacket: Teaching General Relativity
without Mathematics

BodyIn 1975 Bohm
remarked that “Our mathematical procedures seem to obscure
our intuitive and imaginative understanding.” For the past
few decades I have been exploring ways to discuss General
Relativity with undergraduate students without using any
mathematics whatsoever. Instead, I have used pictures,
words, analogies, and anything else I could think of. These
discussions have been with both Physics majors and
non-science liberal arts students. As is common, although I
believe many of my students have learned a great deal, I
have certainly learned much more by forcing myself to
re-think this topic in a non-mathematical way.

BodyIn
thinking about the resources that undergraduate students’
could bring to a course on general relativity (GR), I will focus on a sound understanding of the special relativistic
physics and mathematics of dynamics and of spacetime
structure.
Students encounter this material in courses on
mechanics and on electromagnetism and in units, or entire
courses, devoted to special relativity (SR). Unfortunately,
there is ample evidence that most students do not develop a
coherent understanding of SR and relativistic dynamics from
traditional instruction. For example, Scherr, Shaffer and
Vokos (2002) and references therein document profound
student difficulties that survive traditional SR
instruction. More informally, John Bell (1987) recounts an
amusing and disturbing anecdote about the difficulties that
his CERN colleagues of the late 1970’s had with a problem
involving two rockets connected by a thread. Matsuda and
Kinoshita (2004) report similar difficulties among their
colleagues.
What accounts for such difficulties? Bell suggests, I
think rightly, that they stem from the failure of
traditional instruction to relate dynamical explanations of
the structure and behavior of physical systems, including
rulers and clocks, to the structure and behavior of physical
systems implied by axiomatic presentations of SR. This
failure deprives students of a valuable resource for
learning, namely, their sense of mechanism, and it leaves
individuals who do master axiomatic SR, like Bell’s
colleagues, struggling to reconcile their dynamical and
kinematical understandings of relativistic physics.
Bell argues that to link these understandings one ought
to analyze the structure and behavior of physical systems
moving through a given inertial coordinate system to obtain
dynamical accounts of kinematical effects like Lorentz
contraction. Since such analyses can be technically
demanding, my poster explores other ways of realizing Bell’s
vision of a linked dynamical/kinematical approach to
learning SR. In particular, I discuss the how linking
students’ dynamical and kinematical understanding of
Galilean relativity can be a major step toward a
correspondingly linked dynamical and kinematical
understanding of SR. Note that innovative introductory
physics curricula like Matter and Interactions,
Chabay and Sherwood (2006), provide the somewhat unusual
conceptual and technical resources that students need in
order to learn SR in this new way. Bohm
remarked that “Our mathematical procedures seem to obscure
our intuitive and imaginative understanding.” For the past
few decades I have been exploring ways to discuss General
Relativity with undergraduate students without using any
mathematics whatsoever. Instead, I have used pictures,
words, analogies, and anything else I could think of. These
discussions have been with both Physics majors and
non-science liberal arts students. As is common, although I
believe many of my students have learned a great deal, I
have certainly learned much more by forcing myself to
re-think this topic in a non-mathematical way.

BodyWe describe the
course content and lessons learned teaching simultaneously
offered courses to undergraduate physics and mathematics
majors. A subset of students took both courses. The general
relativity course was offered in the physics curriculum and
focused more on the physics with standard mathematics
prerequisites. The differential geometry course aimed at the
geometry of curves and surfaces ending with a study of
Cartan's equations and applications to computing curvatures
in general relativity.

BodyThis poster
describes how the American Association of Physics Teachers
is adopting some Gordon Research
Conference practices for workshops that AAPT is organizing
to be of particular interest to physics faculty in
universities and colleges. The poster
also points out features of the AAPT workshops that are
quite different from those of a GRC. The inaugural AAPT
topical workshop is an intensive two-day consideration of
"Teaching General Relativity to Undergraduates" at Syracuse
University on July 20 and 21, 2006. Supported by LIGO, AAPT,
The Center for Gravitational Wave Physics at Penn State, and
by the Syracuse Department of Physics, this workshop is
bringing together fifty faculty who either have been
teaching or aspire to teach GR to undergraduates. The
participants will share their wide range of views of how
this can best be done, and they will develop three model
syllabi - one to teach GR with its full mathematical
apparatus; one in which the solutions to Einstein's
equations are taken as given and their physical significance
is explored in depth; and one to convey important concepts
of GR to students who want only a general acquaintance with
the subject.

BodyAs in many
undergraduate programs, the typical physics major at SUNY
Geneseo graduates without much exposure to gravitational
physics beyond Newton's law of gravitation and Kepler's
laws. Students read about topics like black holes and
cosmology mainly through popular expositions. In the
last few years, however, we have seen a growing interest in
and enthusiasm for learning more about gravity. This
was clearly evident in an experimental course offering
titled "GRAVITY: An Introduction to Einstein's General
Relativity" in spring 2006. We will present some ideas
for course organization and teaching techniques, and student
responses to an end-of-semester feedback survey. We
will also present our experience with involving
undergraduates in research in general relativity at SUNY
Geneseo.

BodyExperimental
results and experimental apparatus are fantastic ways to
connect an abstract theory like general relativity with the
tangible and physical with which many students are
interested and familiar. However, undergraduate students
are likely not prepared to appreciate either results or
apparatus without considerable assistance. Some textbooks
may provide adequate explication. If the text does not,
then either the professor will need to select some
experiments to discuss in considerable detail, or the
professor may assign learning about an experiment to each
student and then allow the students to teach one another.
Professors who choose to teach undergraduates about
experiments must bear in mind that these students are likely
to have surprisingly weak understandings of apparatus and
surprisingly little experience and ability to interpret
graphs. General relativity courses offer opportunities for
faculty to introduce students to diverse apparatus and to
enhance students’ abilities to read and interpret graphs.

BodyWhile many
calculations in general relativity require intimate
knowledge of the theory, there are a large number of
problems which can be reduced to calculations which exploit
physics and skills that undergraduates possess or are in the
process of developing. These types of problems provide
excellent opportunities to reinforce early lessons in
fundamental physics by giving students an opportunity to
apply their knowledge and practice their skills against the
exciting backdrop of a modern and evolving branch of
physics. They also provide well defined problems for
exercising and developing skills useful in later research
endeavors (e.g., numerical programming). This poster
describes the elements of undergraduate training which we
have found can be tapped and applied to good effect in
undergraduate GR research projects. Several real life
examples which have led to publications in peer reviewed
research journals are described to illustrate the basic
philosophy we advocate.

BodyOn alternate
years since 1989, I have taught GR for physics majors at SNU.
Prerequisites include Special Relativity, the inertia
tensor, Poisson’s equation, partial derivatives. From the
Principle of Equivalence and Einstein’s interpretation of
gravity as curvature, with the introduction of tensors and
covariant derivatives we develop the Principle of General
Covariance, applied at once to the freely-falling particle.
We next develop Einstein’s Field Equations, derive the
Schwarzschild metric, and calculate the deflection of
starlight and precession of perihelia. We also explore the
Friedmann-Robertson-Walker
metric of cosmology. Today our GR program includes a
three-course sequence: Special Relativity, Black Holes
Seminar, and PHYS 4311-2, General Relativity. Local
research experience includes student-authored papers in AJP
and JURP. One SNU student holds an internship at Stanford
University this summer with Gravity Probe B. Another GR
teaching resource may be mentioned: articles published in
the SPS Observer by the Society of Physics Students.

BodyObservational
astronomy stands at the threshold of an era where
gravitational wave detectors are a tool that regularly
contributes important information to the growing body of
astrophysical knowledge. Detectors such as LIGO and LISA
will probe different regimes of the gravitational wave
spectrum and observe sources that radiate at different
gravitational wavelengths. Unlike their cousins, traditional
electromagnetic telescopes, gravitational wave detectors are
not imaging instruments. How then does a gravitational wave
astronomer take the output from a detector and extract
astrophysical information about the emitting sources? In
this poster we introduce an activity in which students
extract astrophysical information from a simulated
gravitational wave signal. The process described mimics the
way true gravitational wave analysis will be handled by
using plots of a pure gravitational wave signal. The
students directly measure the properties of the simulated
signal, and use the measurements to evaluate standard
formulae for astrophysical source parameters.

BodyWe present the
foundations for a unified treatment of three planar
geometries used in physics: Euclidean space, Galilean
spacetime, and Minkowskian spacetime. Following I.M. Yaglom,
using techniques familiar from the analytic geometry and
trigonometry of Euclidean space, we develop the
corresponding analogues for Galilean and Minkowskian
spacetimes and suggest extensions to the constant-curvature
de-Sitter space-times. We comment on how this unified
treatment suggests a faithful visualization of tensors and
their algebra [consistent with the pictorial representations
in Schouten, Misner-Thorne-Wheeler, and Burke]. We feel that
this provides a new geometric framework for teaching
relativity.

BodyUnderstanding
the twin paradox and resolving it thoroughly is an important
first step for students beginning the study of general
relativity. The complete presentation of this topic given
here is designed around a thought experiment that involves
an easily visualized acceleration and deceleration of one of
the twins relative to an inertial frame such that the twins
are at rest together in the same inertial frame at start and
finish. It shows the calculations of the time that has
passed for each twin according to the one remaining in the
inertial frame and according to the one who travels. Both
twins agree that the one who traveled is younger at finish.
Motional and gravitational time dilation effects interplay
to bring about this agreement. A merry-go-round whose
motion is governed by an explicit formula is used so that
values of velocity, acceleration, gravitational potential,
and proper time can be compared graphically at all places
throughout the trip.

BodyAn interactive
JAVA program plots orbits of test particles and light
flashes in the equatorial plane of a non-spinning
(Schwarzschild) and a variable-spin (Kerr) black hole. The
software displays either the time-development of an orbit or
the entire orbit over extended time. In the latter case the
orbit changes instantly and continuously as the operator
varies initial conditions. For the spinning Kerr black hole,
the display shows the ergosphere (in which no particle can
remain at rest) as well as the outer horizon and inner
(Cauchy) horizon. The operator can use alternative global
coordinate systems appropriate to the given black hole:
Schwarzschild, Boyer-Lindquist, Gullstrand-Panlevé, and
Doran. The interactive time-dependent display complements
the static, analytic presentation of textbooks.